Harville, Discounted Harville and Each Way bets

I have used a chrome extension to calculate the over rounds and the place book over rounds and add to the oddschecker website.

For Top 2 places, the overround should sum to 200%. 

191% place overround means that the place book is over broke, i.e. if the bookie were to take proportioned money on all outcomes then for every £191 in stakes it took it would pay out £200 forcing it to lose money. That's bad for the bookie, but it is good for you, you can take a plus EV bet on the Top 2 odds. The catch is you have to take a negative EV bet on the win odds which can completely cancel out the value of the Top 2 bet. 

Sometimes we can take advantage of an over broke place book. To do so we can calculate the odds the bookies are offering on win and place (easy) and then we need an estimate of the true odds for win and place. Sometimes its relatively easy to get an estimate for win and place when there is a liquid popular market available on betfair exchange. In the case of the Premier League outright this is true for the Winner market, but very less true for the Top 2 market. 

Can we use the Winner market to estimate the true odds for Top 2? One thought experiment goes like this, we have a bowl of 100 balls with the names of the Teams in each ball - 68 balls with Man City, 24 balls with Liverpool etc. These match the win probabilities. If you draw one ball at random then that will simulate precisely the chance of each Team winning the Premier League. Say you draw Man City as winner then since Man City cannot come 1st and 2nd, we remove the rest of the Man City balls from the bowl. This leaves 32 balls, 24 of which are Liverpool. This give an estimate of 24/32 or 75% chance Liverpool will be 2nd (if Man City win). Or 68% x 75% = 51% chance of Man City - Liverpool straight forecast. This a starting point to calculating the probablity Liverpool is 2nd (it will be the sum of all the straight forecast where Liverpool finished 2nd). From there we can get the probability of Liverpool finishing Top 2, simply the probability they are 1st plus probability they are 2nd.

Probability[Man City 1st, Liverpool 2nd] = P[Man City 1st] * P[Liverpool 1st]/(SUM(P[Liverpool 1st] + P[Arsenal 1st] + ... + P[Newcastle 1st])

This is called the Harville formula and it turns out is generally not accurate enough to be useful. It will overestimate the probabilies of the favourites to finish in the place positions. This formula can be improved using the Discounted Harville formula which uses constants lambda and rho to attempt to deal with the bias. The second constant rho is only needed for calculating trifecta's or positions down to 3rd place. We don't need that in this case so we can greatly simplify things. So if Harville is described in the previous formula, Discounted Harville is described below.

Probability[Man City 1st, Liverpool 2nd] = P[Man City 1st] * (P[Liverpool 1st] ^ lambda)/(SUM(P[Liverpool 1st] ^ lambda + P[Arsenal 1st] ^ lambda + ... + P[Newcastle 1st] ^ lambda)

The value of lambda is unknown. It varies depending on the competition, the make up of the event, the varied strengths/weaknesses of the participants. If you set it to 1, it simplifies to the standard Harville formula. In the original paper Henery estimated lambda = 0.76. That paper was more concerned with horse racing in Japan, so it really leaves us completely in the dark as to what value to use for the Premier League. I have estimated different values of lambda in the table below to try to "fit" the probabilities with the range of values that look possible on the betfair exchange. It was already mentioned that there is low liquidity and volume on the Top 2 market, but we can still use it as a guide or limit. For example you can back Man City at 1.11 or lay them at 1.16, so it is likely that the true odds lies somewhere between.

I've only included the lay odds for the win as by and large its a good estimate on its own for a highly liquid, large volume matched market. For the Top 2 (place) exchange odds there is bigger gaps, so I've included both available back and lay (at time of writing).

You can see that for lambda = 1 (which simplifies to original Harville method) the estimate for Top 2 for the favourites has been much shorter than the possible values from the exchange. This was already mentioned.  The value that actually fits best is 0.76, which is Henery's original value based on Horse Racing. For this value most of the probabilities fall in the range that the Top 2 market on the betfair exchange expects are possible.  It still slightly over estimates for Man City, giving 1.08 in decimal odds to finish Top 2. If this indeed were accurate then the market would move to reflect this.  

In conclusion you can see that the Discounted Harville can give decent approximations to the true odds but perhaps not consistent or accurate enough to be of great value. It was originally applied to Horse Racing, so Football might not be a relevant application. 

However it does do well enough to highlight potential value:

• Man City is slightly +EV but its been already noted that even Discounted Harville is supplying an overestimation.
• Liverpool is also small +EV and the exchange is broadly in agreement with the Discounted Harville numbers, the edge is small though.
• Leicester is probably the best candidate for an each way bet. The win odds are close enough to the exchange that you are not losing too bad on the win part of the bet. You are getting 100/1 for Top 2 on the place part, which our process has estimated at about 58.66. Will this bet win? Almost definitely not. Is it +EV? Yes, I think so.
• Everton is also showing as +EV but again small, and this time big gaps in the exchange gives us little confidence in accepting the Harville approximation.

There is no doubt that Each Way backing can sometimes throw up some value on the place part. Some bookies have already removed each way terms on the Premier League market which is an admission that these bookies think that the way the favourites have started the season has left them potentially exposed to each way thieving. In my opinion there is possibly some value still available on those that have chosen not to remove each way terms as yet.  Some of the oddschecker odds are out of date, for example if you can get 1/3 each way terms on BetVictor as oddschecker is showing then this would be great. However I cant find it.